Integrand size = 29, antiderivative size = 429 \[ \int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=-\frac {i (a-i b)^{3/2} (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {i (a+i b)^{3/2} (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{8 b^{3/2} \sqrt {d} f}+\frac {\left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b f}+\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 b f}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f} \] Output:
-I*(a-I*b)^(3/2)*(c-I*d)^(5/2)*arctanh((c-I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2 )/(a-I*b)^(1/2)/(c+d*tan(f*x+e))^(1/2))/f+I*(a+I*b)^(3/2)*(c+I*d)^(5/2)*ar ctanh((c+I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a+I*b)^(1/2)/(c+d*tan(f*x+e))^ (1/2))/f+1/8*(15*a^2*b*c*d^2-a^3*d^3+3*a*b^2*d*(15*c^2-8*d^2)+5*b^3*(c^3-8 *c*d^2))*arctanh(d^(1/2)*(a+b*tan(f*x+e))^(1/2)/b^(1/2)/(c+d*tan(f*x+e))^( 1/2))/b^(3/2)/d^(1/2)/f+1/8*(14*a*b*c*d-a^2*d^2+b^2*(11*c^2-8*d^2))*(a+b*t an(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(1/2)/b/f+1/12*d*(-a*d+13*b*c)*(a+b*tan( f*x+e))^(3/2)*(c+d*tan(f*x+e))^(1/2)/b/f+1/3*d^2*(a+b*tan(f*x+e))^(5/2)*(c +d*tan(f*x+e))^(1/2)/b/f
Time = 7.42 (sec) , antiderivative size = 773, normalized size of antiderivative = 1.80 \[ \int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}+\frac {\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{4 f}+\frac {\frac {3 d \left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 f}+\frac {-\frac {6 b d^2 \left (\sqrt {-b^2} \left (b^2 c \left (c^2-3 d^2\right )-a^2 \left (c^3-3 c d^2\right )+a b \left (6 c^2 d-2 d^3\right )\right )-b \left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {-c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {-a+\sqrt {-b^2}} \sqrt {-c+\frac {\sqrt {-b^2} d}{b}}}-\frac {6 b d^2 \left (\sqrt {-b^2} \left (b^2 c \left (c^2-3 d^2\right )-a^2 \left (c^3-3 c d^2\right )+a b \left (6 c^2 d-2 d^3\right )\right )+b \left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+\frac {\sqrt {-b^2} d}{b}}}+\frac {3 \sqrt {b} d^{3/2} \sqrt {c-\frac {a d}{b}} \left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\right )\right ) \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c-\frac {a d}{b}}}\right ) \sqrt {\frac {b c+b d \tan (e+f x)}{b c-a d}}}{4 \sqrt {c+d \tan (e+f x)}}}{b d f}}{2 d}}{3 b} \] Input:
Integrate[(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(5/2),x]
Output:
(d^2*(a + b*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]])/(3*b*f) + ((d*(1 3*b*c - a*d)*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]])/(4*f) + ((3*d*(14*a*b*c*d - a^2*d^2 + b^2*(11*c^2 - 8*d^2))*Sqrt[a + b*Tan[e + f*x ]]*Sqrt[c + d*Tan[e + f*x]])/(4*f) + ((-6*b*d^2*(Sqrt[-b^2]*(b^2*c*(c^2 - 3*d^2) - a^2*(c^3 - 3*c*d^2) + a*b*(6*c^2*d - 2*d^3)) - b*(2*a*b*c*(c^2 - 3*d^2) - b^2*d*(3*c^2 - d^2) + a^2*(3*c^2*d - d^3)))*ArcTanh[(Sqrt[-c + (S qrt[-b^2]*d)/b]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[-c + (Sqrt[-b^2]*d)/b]) - ( 6*b*d^2*(Sqrt[-b^2]*(b^2*c*(c^2 - 3*d^2) - a^2*(c^3 - 3*c*d^2) + a*b*(6*c^ 2*d - 2*d^3)) + b*(2*a*b*c*(c^2 - 3*d^2) - b^2*d*(3*c^2 - d^2) + a^2*(3*c^ 2*d - d^3)))*ArcTanh[(Sqrt[c + (Sqrt[-b^2]*d)/b]*Sqrt[a + b*Tan[e + f*x]]) /(Sqrt[a + Sqrt[-b^2]]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + Sqrt[-b^2]]*S qrt[c + (Sqrt[-b^2]*d)/b]) + (3*Sqrt[b]*d^(3/2)*Sqrt[c - (a*d)/b]*(15*a^2* b*c*d^2 - a^3*d^3 + 3*a*b^2*d*(15*c^2 - 8*d^2) + 5*b^3*(c^3 - 8*c*d^2))*Ar cSinh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c - (a*d)/b])]*Sqrt [(b*c + b*d*Tan[e + f*x])/(b*c - a*d)])/(4*Sqrt[c + d*Tan[e + f*x]]))/(b*d *f))/(2*d))/(3*b)
Time = 2.93 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {3042, 4049, 27, 3042, 4130, 27, 3042, 4130, 27, 3042, 4138, 2348, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}dx\) |
| \(\Big \downarrow \) 4049 |
| \(\displaystyle \frac {\int \frac {(a+b \tan (e+f x))^{3/2} \left (6 b c^3-5 b d^2 c-a d^3+d^2 (13 b c-a d) \tan ^2(e+f x)+6 b d \left (3 c^2-d^2\right ) \tan (e+f x)\right )}{2 \sqrt {c+d \tan (e+f x)}}dx}{3 b}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \tan (e+f x))^{3/2} \left (6 b c^3-5 b d^2 c-a d^3+d^2 (13 b c-a d) \tan ^2(e+f x)+6 b d \left (3 c^2-d^2\right ) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{6 b}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a+b \tan (e+f x))^{3/2} \left (6 b c^3-5 b d^2 c-a d^3+d^2 (13 b c-a d) \tan (e+f x)^2+6 b d \left (3 c^2-d^2\right ) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{6 b}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {\int -\frac {3 \sqrt {a+b \tan (e+f x)} \left (-d^2 \left (\left (11 c^2-8 d^2\right ) b^2+14 a c d b-a^2 d^2\right ) \tan ^2(e+f x)-8 b d \left (b c^3+3 a d c^2-3 b d^2 c-a d^3\right ) \tan (e+f x)+d \left (a^2 d^3+13 b^2 c^2 d-a b \left (8 c^3-10 c d^2\right )\right )\right )}{2 \sqrt {c+d \tan (e+f x)}}dx}{2 d}+\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 f}}{6 b}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 f}-\frac {3 \int \frac {\sqrt {a+b \tan (e+f x)} \left (-d^2 \left (\left (11 c^2-8 d^2\right ) b^2+14 a c d b-a^2 d^2\right ) \tan ^2(e+f x)-8 b d \left (b c^3+3 a d c^2-3 b d^2 c-a d^3\right ) \tan (e+f x)+d \left (a^2 d^3+13 b^2 c^2 d-a b \left (8 c^3-10 c d^2\right )\right )\right )}{\sqrt {c+d \tan (e+f x)}}dx}{4 d}}{6 b}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 f}-\frac {3 \int \frac {\sqrt {a+b \tan (e+f x)} \left (-d^2 \left (\left (11 c^2-8 d^2\right ) b^2+14 a c d b-a^2 d^2\right ) \tan (e+f x)^2-8 b d \left (b c^3+3 a d c^2-3 b d^2 c-a d^3\right ) \tan (e+f x)+d \left (a^2 d^3+13 b^2 c^2 d-a b \left (8 c^3-10 c d^2\right )\right )\right )}{\sqrt {c+d \tan (e+f x)}}dx}{4 d}}{6 b}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 f}-\frac {3 \left (\frac {\int \frac {-\left (\left (5 \left (c^3-8 c d^2\right ) b^3+3 a d \left (15 c^2-8 d^2\right ) b^2+15 a^2 c d^2 b-a^3 d^3\right ) \tan ^2(e+f x) d^2\right )+\left (c \left (11 c^2-8 d^2\right ) b^3+a d \left (51 c^2-8 d^2\right ) b^2-a^2 \left (16 c^3-33 c d^2\right ) b+a^3 d^3\right ) d^2-16 b \left (\left (3 c^2 d-d^3\right ) a^2+2 b c \left (c^2-3 d^2\right ) a-b^2 d \left (3 c^2-d^2\right )\right ) \tan (e+f x) d^2}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{d}-\frac {d \left (-a^2 d^2+14 a b c d+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 d}}{6 b}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 f}-\frac {3 \left (\frac {\int \frac {-\left (\left (5 \left (c^3-8 c d^2\right ) b^3+3 a d \left (15 c^2-8 d^2\right ) b^2+15 a^2 c d^2 b-a^3 d^3\right ) \tan ^2(e+f x) d^2\right )+\left (c \left (11 c^2-8 d^2\right ) b^3+a d \left (51 c^2-8 d^2\right ) b^2-a^2 \left (16 c^3-33 c d^2\right ) b+a^3 d^3\right ) d^2-16 b \left (\left (3 c^2 d-d^3\right ) a^2+2 b c \left (c^2-3 d^2\right ) a-b^2 d \left (3 c^2-d^2\right )\right ) \tan (e+f x) d^2}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 d}-\frac {d \left (-a^2 d^2+14 a b c d+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 d}}{6 b}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 f}-\frac {3 \left (\frac {\int \frac {-\left (\left (5 \left (c^3-8 c d^2\right ) b^3+3 a d \left (15 c^2-8 d^2\right ) b^2+15 a^2 c d^2 b-a^3 d^3\right ) \tan (e+f x)^2 d^2\right )+\left (c \left (11 c^2-8 d^2\right ) b^3+a d \left (51 c^2-8 d^2\right ) b^2-a^2 \left (16 c^3-33 c d^2\right ) b+a^3 d^3\right ) d^2-16 b \left (\left (3 c^2 d-d^3\right ) a^2+2 b c \left (c^2-3 d^2\right ) a-b^2 d \left (3 c^2-d^2\right )\right ) \tan (e+f x) d^2}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 d}-\frac {d \left (-a^2 d^2+14 a b c d+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 d}}{6 b}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 4138 |
| \(\displaystyle \frac {\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 f}-\frac {3 \left (\frac {\int \frac {-\left (\left (5 \left (c^3-8 c d^2\right ) b^3+3 a d \left (15 c^2-8 d^2\right ) b^2+15 a^2 c d^2 b-a^3 d^3\right ) \tan ^2(e+f x) d^2\right )+\left (c \left (11 c^2-8 d^2\right ) b^3+a d \left (51 c^2-8 d^2\right ) b^2-a^2 \left (16 c^3-33 c d^2\right ) b+a^3 d^3\right ) d^2-16 b \left (\left (3 c^2 d-d^3\right ) a^2+2 b c \left (c^2-3 d^2\right ) a-b^2 d \left (3 c^2-d^2\right )\right ) \tan (e+f x) d^2}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{2 d f}-\frac {d \left (-a^2 d^2+14 a b c d+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 d}}{6 b}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 2348 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}+\frac {\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 f}-\frac {3 \left (-\frac {d \left (-a^2 d^2+14 a b c d+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+\frac {\int \left (\frac {\left (-5 \left (c^3-8 c d^2\right ) b^3-a \left (45 c^2 d-24 d^3\right ) b^2-15 a^2 c d^2 b+a^3 d^3\right ) d^2}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {16 b^3 d^5-16 a^2 b d^5-96 a b^2 c d^4-48 b^3 c^2 d^3+48 a^2 b c^2 d^3+32 a b^2 c^3 d^2+i \left (-32 a b^2 d^5-48 b^3 c d^4+48 a^2 b c d^4+96 a b^2 c^2 d^3+16 b^3 c^3 d^2-16 a^2 b c^3 d^2\right )}{2 (i-\tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {-16 b^3 d^5+16 a^2 b d^5+96 a b^2 c d^4+48 b^3 c^2 d^3-48 a^2 b c^2 d^3-32 a b^2 c^3 d^2+i \left (-32 a b^2 d^5-48 b^3 c d^4+48 a^2 b c d^4+96 a b^2 c^2 d^3+16 b^3 c^3 d^2-16 a^2 b c^3 d^2\right )}{2 (\tan (e+f x)+i) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}\right )d\tan (e+f x)}{2 d f}\right )}{4 d}}{6 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}+\frac {\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 f}-\frac {3 \left (-\frac {d \left (-a^2 d^2+14 a b c d+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+\frac {-\frac {2 d^{3/2} \left (-a^3 d^3+15 a^2 b c d^2+a b^2 \left (45 c^2 d-24 d^3\right )+5 b^3 \left (c^3-8 c d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {b}}+16 i b d^2 (a-i b)^{3/2} (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )-16 i b d^2 (a+i b)^{3/2} (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{2 d f}\right )}{4 d}}{6 b}\) |
Input:
Int[(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(5/2),x]
Output:
(d^2*(a + b*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]])/(3*b*f) + ((d*(1 3*b*c - a*d)*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]])/(2*f) - (3*(((16*I)*(a - I*b)^(3/2)*b*(c - I*d)^(5/2)*d^2*ArcTanh[(Sqrt[c - I*d]*S qrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])] - (16*I )*(a + I*b)^(3/2)*b*(c + I*d)^(5/2)*d^2*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b* Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])] - (2*d^(3/2)*(15* a^2*b*c*d^2 - a^3*d^3 + 5*b^3*(c^3 - 8*c*d^2) + a*b^2*(45*c^2*d - 24*d^3)) *ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f* x]])])/Sqrt[b])/(2*d*f) - (d*(14*a*b*c*d - a^2*d^2 + b^2*(11*c^2 - 8*d^2)) *Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/f))/(4*d))/(6*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. )*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(c + d*x)^m*(e + f*x)^ n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[P x, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0])) && !(IGtQ[m, 0] && IGtQ[n, 0])
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2 , 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || I ntegerQ[m]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) )
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_. ) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C *(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[ c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, S imp[ff/f Subst[Int[(a + b*ff*x)^m*(c + d*ff*x)^n*((A + B*ff*x + C*ff^2*x^ 2)/(1 + ff^2*x^2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f , A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Timed out.
\[\int \left (a +b \tan \left (f x +e \right )\right )^{\frac {3}{2}} \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}d x\]
Input:
int((a+b*tan(f*x+e))^(3/2)*(c+d*tan(f*x+e))^(5/2),x)
Output:
int((a+b*tan(f*x+e))^(3/2)*(c+d*tan(f*x+e))^(5/2),x)
Leaf count of result is larger than twice the leaf count of optimal. 15404 vs. \(2 (353) = 706\).
Time = 18.72 (sec) , antiderivative size = 30824, normalized size of antiderivative = 71.85 \[ \int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=\text {Too large to display} \] Input:
integrate((a+b*tan(f*x+e))^(3/2)*(c+d*tan(f*x+e))^(5/2),x, algorithm="fric as")
Output:
Too large to include
Timed out. \[ \int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate((a+b*tan(f*x+e))**(3/2)*(c+d*tan(f*x+e))**(5/2),x)
Output:
Timed out
\[ \int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=\int { {\left (b \tan \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*tan(f*x+e))^(3/2)*(c+d*tan(f*x+e))^(5/2),x, algorithm="maxi ma")
Output:
integrate((b*tan(f*x + e) + a)^(3/2)*(d*tan(f*x + e) + c)^(5/2), x)
\[ \int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=\int { {\left (b \tan \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*tan(f*x+e))^(3/2)*(c+d*tan(f*x+e))^(5/2),x, algorithm="giac ")
Output:
integrate((b*tan(f*x + e) + a)^(3/2)*(d*tan(f*x + e) + c)^(5/2), x)
Timed out. \[ \int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=\int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2} \,d x \] Input:
int((a + b*tan(e + f*x))^(3/2)*(c + d*tan(e + f*x))^(5/2),x)
Output:
int((a + b*tan(e + f*x))^(3/2)*(c + d*tan(e + f*x))^(5/2), x)
\[ \int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=\left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}\, \tan \left (f x +e \right )^{3}d x \right ) b \,d^{2}+\left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}\, \tan \left (f x +e \right )^{2}d x \right ) a \,d^{2}+2 \left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}\, \tan \left (f x +e \right )^{2}d x \right ) b c d +2 \left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}\, \tan \left (f x +e \right )d x \right ) a c d +\left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}\, \tan \left (f x +e \right )d x \right ) b \,c^{2}+\left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}d x \right ) a \,c^{2} \] Input:
int((a+b*tan(f*x+e))^(3/2)*(c+d*tan(f*x+e))^(5/2),x)
Output:
int(sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**3,x)*b *d**2 + int(sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x) **2,x)*a*d**2 + 2*int(sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*ta n(e + f*x)**2,x)*b*c*d + 2*int(sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)* b + a)*tan(e + f*x),x)*a*c*d + int(sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f *x)*b + a)*tan(e + f*x),x)*b*c**2 + int(sqrt(tan(e + f*x)*d + c)*sqrt(tan( e + f*x)*b + a),x)*a*c**2